What is the difference between model and data distributions?
Yes. In Machine Learning we consider that the samples in your training set are sampled from an underlying distribution called the data generating distribution.

Generative models classify the samples by trying to learn the distribution of the data. In most cases, either the model is incapable of doing so, or the training samples aren't enough to properly describe the data-generating distribution, so the model learns an approximation of this. This is what you call the model's distribution.

You can find more info about these concepts in a more detailed answer I wrote. If you're familiar with GANs, you can also read this post, to see where these two concepts come into play when training the two networks.